Strong shift equivalence and algebraic K-theory

Mike Boyle; Scott Schmieding

doi:10.1515/crelle-2016-0056

Article

Strong shift equivalence and algebraic K-theory

Mike Boyle and Scott Schmieding

Published/Copyright: December 1, 2016

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2019 Issue 752

Abstract

For a semiring ℛ, the relations of shift equivalence over ℛ (SE-⁢ℛ) and strong shift equivalence over ℛ (SSE-⁢ℛ) are natural equivalence relations on square matrices over ℛ, important for symbolic dynamics. When ℛ is a ring, we prove that the refinement of SE-⁢ℛ by SSE-⁢ℛ, in the SE-⁢ℛ class of a matrix A, is classified by the quotient N⁢K1⁢(ℛ)/E⁢(A,ℛ) of the algebraic K-theory group N⁢K1⁢(ℛ). Here, E⁢(A,ℛ) is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over ℛ that the refinement of its SE-⁢ℛ class into SSE-⁢ℛ classes corresponds precisely to the refinement of the GL⁢(ℛ⁢[t]) equivalence class of I-t⁢A into El⁢(ℛ⁢[t]) equivalence classes. We then show this refinement is in bijective correspondence with N⁢K1⁢(ℛ)/E⁢(A,ℛ). For a general ring ℛ and A invertible, the proof that E⁢(A,ℛ) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For ℛ commutative, we show ∪AE⁢(A,ℛ)=N⁢S⁢K1⁢(ℛ); the proof rests on Nenashev’s presentation of K1 of an exact category.

Funding source: Danmarks Grundforskningsfond

Award Identifier / Grant number: DNRF92

Funding source: Natural Sciences and Engineering Research Council of Canada

Award Identifier / Grant number: Discovery grants of David Handelman and of Thierry Giordano at the University of Ottawa

Funding statement: Mike Boyle was supported by the Danish National Research Foundation, through the Centre for Symmetry and Deformation (DNRF92), and by the NSERC Discovery grants of David Handelman and of Thierry Giordano, at the University of Ottawa.

Acknowledgements

We thank Jonathan Rosenberg for all the K-theory education and consultation. We thank Wolfgang Steimle for the content of Remark 4.9 and we thank David Handelman for completing the proof of Proposition 4.10. We are grateful to Andrew Ranicki, Jonathan Rosenberg and Charles Weibel for their books [29, 30, 41], without which we might not have written this paper. Finally, we are grateful to the referee for a very detailed and thorough review, which has improved the presentation and accuracy of the paper.

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Received: 2015-06-16

Revised: 2016-08-12

Published Online: 2016-12-01

Published in Print: 2019-07-01

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https://doi.org/10.1515/crelle-2016-0056